LOCA is a new software library for performing bifurcation analysis of large-scale applications. The source code has been available for download since April 19, 2002. The algorithms in LOCA are being designed as part of an ongoing research project at Sandia National Laboratories in Albuquerque into scalable stability analysis algorithms. When implemented with an application code, LOCA enables the tracking of solution branches as a function of system parameters and the direct tracking of bifurcation points. LOCA (which is written in ”C”) is designed to drive application codes that use Newton’s method to locate steady-state solutions to nonlinear problems. The algorithms are chosen to work for large problems, such as those that arise from discretizations of partial differential equations, and to run on distributed memory parallel machines. The approach in LOCA for locating and tracking bifurcations begins with augmenting the residual equations defining a steady state with additional equations that describe the bifurcation. A Newton method is then formulated for this augmented system; however, instead of loading up the Jacobian matrix for the entire augmented system (a task that involved second derivatives and dense matrix rows), bordering algorithms are used to decompose the linear solve into several solves with smaller matrices. Almost all of the algorithms just require multiple solves of the Jacobian matrix for the steady state problem to calculate the Newton updates for the augmented system. This greatly simplifies the implementation, since this is the same linear system that an application code using Newton’s method will already have invested in. Only the Hopf tracking algorithm requires the solution of a larger matrix, which is the complex matrix involving the Jacobian matrix and an imaginary multiple of the mass matrix.

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  1. Dhillon, Daljit Singh J.; Milinkovitch, Michel C.; Zwicker, Matthias: Bifurcation analysis of reaction diffusion systems on arbitrary surfaces (2017)
  2. Net, M.; Sánchez, J.: Continuation of bifurcations of periodic orbits for large-scale systems (2015)
  3. Sharma, Sanjiv; Coetzee, Etienne B.; Lowenberg, Mark H.; Neild, Simon A.; Krauskopf, Bernd: Numerical continuation and bifurcation analysis in aircraft design: an industrial perspective (2015)
  4. Thompson, J.Michael T.; Sieber, Jan: Predicting climate tipping as a noisy bifurcation: a review (2011)
  5. Ciani, A.; Kreutner, W.; Frouzakis, C.E.; Lust, K.; Coppola, G.; Boulouchos, K.: An experimental and numerical study of the structure and stability of laminar opposed-jet flows (2010)
  6. Sánchez, Juan; Net, Marta: On the multiple shooting continuation of periodic orbits by Newton-Krylov methods (2010)
  7. Valli, A.M.P.; Elias, R.N.; Carey, G.F.; Coutinho, A.L.G.A.: PID adaptive control of incremental and arclength continuation in nonlinear applications (2009)
  8. Bader, Brett W.; Schnabel, Robert B.: On the performance of tensor methods for solving ill-conditioned problems (2007)
  9. Juncu, Gheorghe: Multiplicity analysis of a nonisothermal finite cylindrical catalyst pellet (2007)
  10. Lasater, M.S.; Kelley, C.T.; Salinger, A.G.; Woolard, D.L.; Recine, G.; Zhao, P.: Analysis of a scalable preconditioner for the Wigner-Poisson equation (2007)
  11. García-Archilla, Bosco; Sánchez, Juan; Simó, Carles: Krylov methods and determinants for detecting bifurcations in one parameter dependent partial differential equations (2006)
  12. Juncu, Gh.; Mosekilde, E.; Popa, C.: Numerical experiments with MG continuation algorithms (2006)
  13. Lasater, M.S.; Kelley, C.T.; Salinger, A.G.; Woolard, D.L.; Zhao, P.: Parallel parameter study of the Wigner-Poisson equations for RTDs (2006)
  14. Pawlowski, R.P.; Salinger, A.G.; Shadid, J.N.; Mountziaris, T.J.: Bifurcation and stability analysis of laminar isothermal counterflowing jets (2006)
  15. Radziunas, M.: Numerical bifurcation analysis of the traveling wave model of multisection semiconductor lasers (2006)
  16. Salinger, A.G.; Burroughs, E.A.; Pawlowski, R.P.; Phipps, E.T.; Romero, L.A.: Bifurcation tracking algorithms and software for large scale applications (2005)
  17. Burroughs, Elizabeth A.; Romero, Louis A.; Lehoucq, Richard B.; Salinger, Andrew G.: Linear stability of flow in a differentially heated cavity via large-scale eigenvalue calculations (2004)
  18. Bou-Rabee, Nawaf M.; Romero, Louis A.; Salinger, Andrew G.: A multiparameter, numerical stability analysis of a standing cantilever conveying fluid (2002)
  19. Salinger, Andrew G.; Lehoucq, Richard B.; Pawlowski, Roger P.; Shadid, John N.: Computational bifurcation and stability studies of the 8:1 thermal cavity problem (2002)